3.2.1 Branch-and-bound algorithms
Most exact global optimization methods are based on a branch-and-bound frame-work. In general, the branch-and-bound algorithm starts by partitioning the feasible region of the problem into two or more sub-regions (branching process), and constructs a relaxation for each sub-region. This yields a lower bound on the global minimum cost, possibly also an upper bound (bounding process). The branching process is then applied recursively, and defines a search tree in which the nodes represents the sub-regions. Nodes in the resulting search tree are pruned when its lower bounds exceed the best upper bound found so far. The algorithm halts when the tree is empty or the best lower and upper bounds are sufficiently close.
3.2.1.1 Primal-dual decomposition methods
In an effort to improve the method described in Section 3.1.4,Visweswaran and Floudas (1990) suggested the first global optimization algorithm based on a de-composition technique and branch-and-bound. The problem is decomposed into primal and dual subproblems to provide upper and lower bounds on the global solution. Gradient information of the Lagrange function is used to partition the
Section 3.2. Exact solution techniques
current domain into sub-domains, and the procedure is repeated until it con-verges to the global solution. Some improvements of this method are observed in e.g. (Visweswaran and Floudas,1993) and (Androukakis et al.,1996).
3.2.1.2 Linear relaxation based algorithms
Linear relaxations of problems involving some bilinear function f (x, y) = xy, where (x, y) ∈ D, are obtained by the convex and concave envelopes (see Section 1.2) of f , denoted vexDf (x, y) and cavDf (x, y), respectively. It can be shown (seeAl-Khayyal and Falk,1983;McCormick,1976) that the convex and concave envelopes of f on the rectangle D = [x, x] × [y, y] are given by, respectively,
vexDf (x, y) = maxyx + xy − xy, yx + xy − xy , (3.1) cavDf (x, y) = minyx + xy − xy, yx + xy − xy . (3.2)
Linear relaxations of the pooling problem formulations given in Section 2.2 are obtained by replacing all occurrences of the bilinear terms by new variables, and by bounding each new variable between its corresponding envelopes. A branch-and-bound algorithm based on such relaxations, where in each iteration the rectangle is divided into four sub-rectangles, was first applied to the pooling problem byFoulds et al.(1992).Audet et al.(2004) suggested a branch-and-cut algorithm, which is an improvement of the above branch-and-bound technique.
The reformulation-linearization technique (RLT) (Sherali and Alameddine, 1992) is a methodology for constructing tight linear relaxations of a noncon-vex problem. The second step, linearization, was already discussed above. The first step, reformulation, is to add new valid constraints obtained by multiplying two original constraints.
It is interesting to note that one can arrive to the McCormick’s convex and concave envelopes (3.1)–(3.2) by applying the RLT to the bound constraints of x and y. For example, multiplying (x − x) ≥ 0 and (y − y) ≥ 0 yields the constraint xy ≤ yx + xy − xy, which is one of the constraints suggested by (3.2).
Quesada and Grossmann(1995) applied the RLT to obtain a relaxation which is used within a spatial branch and bound algorithm that uses a nonlinear solver
to provide upper bounds. The result in several instances showed that a few branch-and-bound nodes were needed to verify the global solutions.
Sahinidis and Tawarmalani(2005) applied their branch-and-reduce algorithm, which uses the McCormick’s relaxation as lower-bounding and local and random search as upper-bounding techniques. In addition, it uses various range reduction techniques. This algorithm is implemented in the generic global optimization code, BARON (Sahinidis, 1996), by use of the PQ-formulation. When applying their code to standard instances from the literature, they were able to reduce the running time and the size of the search tree significantly.
Liberti and Pantelides (2006) proposed an improved relaxation technique re-ferred to as the reduced reformulation linearization technique (RRLT), and in-corporated it in a spatial branch-and-bound algorithm. The authors also sug-gested an algorithm that automatically constructs this relaxation for large and sparse NLPs such as the pooling problem. They applied their algorithm to com-mon pooling instances where the results showed that tight linear relaxations and hence faster convergence are provided.
Piecewise-linear relaxations have been proposed by Wicaksono and Karimi (2008) andGounaris et al.(2009), who utilize piecewise linearization schemes by partitioning the original domain of the variables involved in the bilinear terms into smaller sub-domains. Applying the McCormick relaxation for each of the result-ing sub-domains, and usresult-ing binary variables to select the optimal sub-domain, resulted in an efficient relaxation that can be used in the branch-and-bound framework to accelerate convergence.
3.2.1.3 Lagrangian relaxation based algorithms
The Lagrangian relaxation is a useful technique when the problem’s constraints can be decomposed into “difficult” and “easy” ones. The difficult constraints are relaxed by adding them to the objective with weight (Lagrange multipliers), and thereby the solution provides a lower bound on the global solution of the original problem. In the pooling problem, the difficult constraints are the bilinear ones.
As shown in Chapter 2, these constraints arise from quality balances around pools and the quality bounds at terminals.
Section 3.2. Exact solution techniques
Ben-Tal et al.(1994) studied the Lagrangian relaxation of their Q-formulation (see Section 2.2.2.1 for details). The associated Lagrangian dual, which gives a lower bound on the minimum cost, is solved by analyzing the simplex n
y ∈ RS×I:P
s∈Ni−ysi = 1o
. This relaxation is integrated in a branch-and-bound algorithm that divides the simplex into smaller ones. Upper branch-and-bounds on the global minimum cost are found by local search.
Adhya et al. (1999) introduced a Lagrangian relaxation by dualizing all the constraints in the P-formulation except for the variable bounds. The solution of the resulting Lagrangian subproblem is approximated by solving a sequence of MILPs. They also proved that the Lagrangian relaxation provides tighter lower bounds than standard linear relaxation does in the case of more than one quality parameter. A similar Lagrangian relaxation was suggested byAlmutairi and Elhedhli(2009).
3.2.2 Semidefinite programming relaxations
In PaperC(Frimannslund et al.,2010), we suggest a technique based on a series of semidefinite programs (or linear matrix inequality (LMI) relaxations) to solve the pooling problem. LMI relaxations are used to turn general (nonconvex) optimization problems, where the objective and the constraints are polynomials, into a sequence of convex positive semidefinite programs (Lasserre,2001a,b).
The general idea of this technique is as follows. Consider the optimization problem,
f∗= min
x∈Rn{f0(x) : x ∈ Ω} , (3.3) and assume for simplicity that Ω is compact and f0is continuous. Then the prob-lem (3.3) can be turned into a convex probprob-lem by minimizing over the set, B(Ω), of all Borel probability measures µ supported on Ω. The resulting optimization problem,
µ∗= min
µ∈B(Ω)
Z
f0(x)dµ,
has the same global optimum value as the original problem. However, finding the probability distribution µ∗ on the support Ω is done by characterizing its moment sequences, which is an infinite-dimensional convex optimization problem
known as the moment problem (Lasserre, 2010). Instead of solving an infinite-dimensional problem, a truncated moment sequences are determined, which can be cast as an LMI relaxation. By increasing the order of the moment sequences, a tighter relaxation is obtained.
By applying the above technique to the pooling problem with a single quality parameter,Frimannslund et al.(2010) show that if the feasible set has a nonempty interior, then we have a finite sequence of LMI relaxations with increasing order that converges to the global optimum. For a fixed relaxation order, this technique thus provides tight lower bounds for the global minimum cost. Based on the experiments, we show that for low order relaxations, the lower bound provided by this technique matches the true global optimum in several small instances.